Remark 2.4 Another interesting \(\ast \) -subalgebra of \({L}^1_B\) , often considered in the
literature (see, e.g., [1]}, [2]}), is made
up of compactly supported continuous functions
\({L}^1_{B,{\rm c}}:={L}^1_B\cap {C}_{c}\big (\mathbb {R}^2\big )\) . The algebraic structure is
preserved by the fact that the magnetic twisted convolution of two compactly supported
continuous functions is again a compactly supported continuous function. This
fact can be proved exactly as in [3]}, where the presence of the 2-cocycle \(\Phi _B\)  is harmless. The density follows from the density of \({C}_{c}\big (\mathbb {R}^2\big )\)  in
\(L^1\big (\mathbb {R}^2\big )\)  [3]}. In the following we will not make
use of this subalgebra.

